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# Introductory Algebra MTH 60

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This 45 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 60 at Portland Community College taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/224642/mth-60-portland-community-college in Math at Portland Community College.

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Date Created: 10/19/15

MTH 60 Sim onds class Key Concepts Introduction to slope MartinGaye sections and practice problems 34 1 93 odd Example 1 For each of figures 14 sToTe The scale on each axis and use The concepT of rise over39 r39un To find The slope of each line xscole yscole slope Figure 1 m Va 4 1 xscole yscole slope bin 4 r I M 395 Figure2 er 4 3 Page 1 of 10 MTH 60 Simonds class x scule y scule slope as 2 S T5 4 M run Figure 3 slope y scule 5 53 SWL 4quot Figure 4 yz yi x2 7x1 Definition The slope ofthe line connectingthe point xvyl and xzyz where yl 5 yZ is m Slope is commonly referred to as rise over runquot where a positive rise means you go up a negative rise means you go down a positive run means you go right and a negative run means you go Ie Page 2 of 10 MTH 6D 7 Stmunus mass Example 2 Find the slope of the line connecting the points 57 3 and 74 2 O ionA O ionB x yt3 x4 yt2 11 IlL Page 3 0H MTH B rswmunds mass Xamg e Fmd m s opz of m m conmmg m pews 127 5 and 7510 Pagezzumu xF397 i A a LI MTH 60 Simonds class 7 439 7 Examgle 5 2 Find 2 poinTs on The line y7x 4 and use Those poinTs To deTermine The slope of The line t Um 9 L l yz LI 0 yl M 91 I x2 0 7 x1 Z x 7 Examgle 6 1 Find 2 poinTs on The line y 79x 7 and use Those poinTs To deTermine The slope of The line L AHAquot Iyi 3 2 1 L yl Yl X 7 39L x O I x1 0 1 1 33 Examgle 7 Find 2 poinTs on The line x 7 2y 7 and use Those poinTs To deTermine The slope of The line cuAJ e f 91 M a yz O I yl X1 7lt 0 7 if a 1 S I 3 Page 5 of 10 MTH 60 Simonds class l l UJ lh J Examgle 8 Find The slope of The lines in figures 7 and 8 LAJL UadlL J JlVQ m XML lxwl39ll Figure 7 Figure 8 HLquot31 4 313 Lr39 yL quot i 7 SJ 7 3 m 7 o 0 Examgle9 5 Draw Two lines onTo Figure 9 One line should pass Through The poinT 0 4 and The oTher should pass Through The poinT 02 BoTh lines should have a slope of 2 WhaT else do The Two lines have in common 439 7 39L l Ahi M l L 13qu 3 41 Rmir I 6 5 4 3 rth m Figure9 fdn Page 6 of 1 Pamllu ll lip P39l39w39l SINquot MTH 60 Sim onds class Example 10 Find The slope of each of The lines in figures 10 and 11 In each case sTaTe which of The Two lines has The greaTer slope Figure 11 Figure 10 a M 2 2 ML M wlL MC 7 my line 5393 1394 39 iebli sl True orThe steeper the line the greater the slope r False The steeper the line the greater the absolute value ofthe slope MA gtmB I AIJ Jh ft I41 Page 7 of 10 MTH 60 r Simonds class Example 11 One of the lines in Figure 12 has a slope of 5 one has a slope of one has a slope of one has a slope ofil and one has a slope of 7 decide which line is which Please note that scales and a grid have deliberately been omitted the purpose of this problem is for you to coma392 the slopes of the lines no calcularethe slopes of the lines B C D Example 12 Graph onto Figure 13 the lines with equations y x 7 4 and y 73x 1 What do you observe about the lines What is the slope of each line How are the slopes arithmetically related quot1 4 3 1 JL 6 75 74 73 72 71 J XH 15 0 l M 3 TVA Pb Ania1 5 013 in 3 shquot 1 Figure 13 Page 8 of 10 71 m 39xgtltJwl 44 le W L OPfBJ39ll L Fr 2 I39Lbl MTH 60 Simonds class Exam9e13 Bernice needed To Touch up her greys Bernice39s sTylisT Bernard convinced her To Try life as a blond for a monTh The bleach Bernard used required Bernice To siT under The hairdryer for several minuTes The graph in Figure B shows The TemperaTure degrees FahrenheiT of Bernice39s hair for The firsT 4 minuTes afTer The dryer was Turned on Find The slope of The line segmenT including uniT and inTerpreT The slope as a raTe of change temperature 0F of Bernice s hair r K 1 4 lt Iquot r J 3 O 9 amount of time min the hairdryer has been engaged URN quc39 rLc J K f kt 3 11 39 4 9 t yl h b BL lg39l A 391 LLcinJ o A 0 4 39I LL Lbnfha fwlt 5 0 uPn39 Example 14 Figure K shows a graph of Mr KiTTy39s weighT lb t weeks afTer The CarTmans broughT him home CalculaTe The slope of The line g segmenT including unf and inTerpreT The a slope as a raTe of change 9 g v lJ a lquot 3 m E 1 IL 2 1 L14an lL lLM 4 Number of weeks in Cartman household Kfl7k ll39l39ler quotl lw Cx l39MMS LnUll Mr K lqML I kg gk39lquot Jalqg ingkF a lquot quotl Q 39 O COAY IJ 4 6P99f10 IL LJIlt MTH 60 Simonds class Examigle 15 Suppose ThaT you have a line where The y coordinaTes are The speed fTs aT which a rolly polly is Travelling and The x coordinaTes are The number of seconds ThaT have passed since T Bone flung The rolly polly inTo The air AfTer 1 second The bug is Travelling aT 49 fTs and afTer 2 seconds iT39s Travelling aT 33 fTs Find The slope of The line including uniTl and inTerpreT The slope as a raTe of change x l in M 1 JA 1111 52 5quot 4 0 L quotT5 n14 39 cf 7 iris P S l JrS quot 1 33 L quot o f O Haggai L A H lly polx3 JLUuJIJ 5 l lu mi thi mi h l ML Examigle 16 The raTe aT which The Murphy39s bed leaks depends upon The ToTal weighT of The people lying upon iT This relaTionship is linear over The inTerva shown in Table 1 Find The slope of This line including uniTl H x II pLb 7 FVA 4 Lquot rile S Iopc NA n I39MA 3 39L rimiv oW T Page 100f10 MTH 60 Simonds class Key Concepts The pointslope form for the equation of the line MartinGave sections and practice problems 36 1 61 odd The answers are written in slopeintercept form on page 6 of this document Pointslope The equation of the line with slope m that passes through the point qy1 can be found using the template y 7 y1 mx 7 x1 Examgle 1 Use point slope to find the equation of the line which passes through the point 27 with a slope of 76 Write the equation in slope intercept form X1 r xquot J v1 quotC7 l M L 3 a 4 Huh 3 ltgtc H v 7 rwm Cw Ln 7 m H7 v Use point slope to find the equation of the line which passes through the points 710 and 9120 Write the equation in slope intercept form 39Il XL L I M L M J I o YL 4 iV xgtlt Loo j 1x 9 quotquot1 x 2 YdI 1 3 L I Jx b 0w L H9 7 Xg i 209 QCQ39Zquot j T39la kinda 5 VI 3 9395 J J L Page lof7 X f yyl X JQIiBwSimunus ciass W x ltgt male 3 Use poinTslope To find The equaTion of The line in Figure 1 WriTe The equaTion in slopeinTercepT form x11 A rug fl 11 F 3391 l h r V nKY X 7 Sr quotz39r 39 ol u K k39 Ij 39qxt LJ S 7quotjxaTA y r 7 Y39 gt O r 5 lx Pit 1 Jr 7 1 1 1 g 74 7 7 1r 3 w swh I Whg would iT be kind of sill To use oinTslo e line in Figure WhaT is The equaTion of The line u L m Jn an wt L 7 P1 H S u 4 439 3115 439 TL uquot 65 j L CLcJk CH3 2 Y ix J 3 a 0 J I Page 2 of 7 MTH 60 Simonds class Examgle 5 Find The equaTion of The line ThaT passes Through The poinTs 37 8 and 312 Il Q I M 3 g PMCM J B4645 X S mm w Example6 Use poinT slope To find The equaTion of The line ThaT passes Through The poinT L7 ThaT is perpendicular To The line wiTh equaTion 73x 7 4y 6 F51 Sh J J7lt fj L 3 2 quotlj3 j v 3x1 quoti s x i L 1 3 j 3 f x 7 g gang LL idph 4 339 3 37 I l 3 itx Lp J 1 7 j quot um In v ZW Page30f7 39 7 7 Ll3 MTH 60 Simonds class Relative steepness When comparing two nonparallel nonvertical lines the steeper of the two lines is the line whose slope has the greater absolute value Examgle 6 In each of figures 3 6 Two lines are shown In each case sTaTe which line has The greaTer slope NoTe I39m noT asking which line39s slope has The greaTer absoluTe value yAlHJ c y B M nl39 X X D A L s Jam JIVL D 5 rck l tr j J Ilerk H E y Ly M F lt gtG X X lt gto 39 F L o J jrwIJquot J39lt l9rQ o39 fvhl Page 4 of 7 MTH 60 Simonds class Example7 739 a o I The purchase of real esTaTe in DeTroiT was noT a good invesTmenT anyTirne beTween 1968 and 2 On January 1 1970 The house aT The corner of Livernois Ave and Tireman STreeT had a value of 143000 Five years aTer The value had fallen To 86000 LeT39s defineg To be The number of years ThaT had passed since January 1 1970 LeT39s also assume ThaT The decline in The value of The house was linear a Find The slope of The line including uniT and inTerpreT The slope as a raTe of change 74 M My M L X J s J39Hjlooaquot 4 moo X YCDOi Il Dvr 5 7r 5 7Ioot S39yr t l 00 l rrlc rl I Lc luth olltcrltgk l 39l Lw Coal A 4 4 ML 9n lyr b Find The equaTion of The line and use iT To predicT when The house39s value had fallen To 50000 7 ML 0 31 3144 xw4 9 C 0439 J LbyJ quot VJ39 7 V36 3 1 on J J 06 ll rn 39 7 LJL sy UL saw M g bJ cab IJkob39y F VBIWO L 1t 0 b 73 000 f0 y llIV l o 3 l y IBM 1 f boloua alt 7 IV393age50f7 A I i 7 MTH 60 Simonds class Exalee 8 If your Taxable income in Oregon is greaTer Than 6500 Then The amounT of income Tax you have To pay for a given year is 403 plus 9 of The amounT of Taxable income you have over 6500 The formula for This Tax is T40309I i6500 where T is The Tax owed by a person whose Taxable income is I dollars A graph of The equaTion is shown in Figure 7 Use The graph To esTimaTe The Taxable income of a person who had To pay 5500 in Oregon sTaTe income Tax for 2005 Then use The formula To find The exacT Taxable income of a person who paid 5500 in Oregon sTaTe income Tax for 2006 t A rllc II LO t J J c 339 con Figure 7 u 95 L Oregon state income tax 8 r6 0 CH3 O f i 466 403F0 lI 5 0 71 0 6 3 0 b o f ch o l Oregon Tax Due 55 1 C3I3333 aol39w lj Taxable Income I 5 on J lt P LyJ l LyaLl i l 3 r33 13 Page 6 of 7 MYH an 7 Smmnds dass Wm Use pumhs upe 1 nd We equmuh uf Me hhe whmh passes Mruugh Me puhvs 53 and 7614 wme Me eqummn h S upermfercepf farm 2 Use pumhs upe m nhdvhe eqummn unhe hhe whmh passes Mruugh Me pmm 735 mm 15 e1su perpehmeumr m We hhe wwh equenuh 6x2y11 wh Weheepv farm He Me equenuh 1h smper 3 The ans uf amgtlt1cc1b Ndes 1h Wesh1hgmhbc 1s dependem r1112 1sy2 rd 1 Wmquot 15 Me s1upe uf M15 hhe hemmhg UHquot Ihvehphev 1he upe es arme ufchange Wmquot 15 Me y mercepf uf M15 hhe hemmhg UHquot Ihvehphev Me yrmmrcepf h 1he euhvexv uf1h1s ques un 12whc11 dues h 1e11 yuu ebum 1he ans uf cab Ndes 1h Wesh1hgmhb C W The phub1em 1s musHy famucnr 121171 r2c1Hy1s fur Me mus pen huw cab fares are devehm med h 1 c 4 Fwd Me equenuh unhe hhe Mm passes Waugh Mepu1m8 727 2 84 1hev 1s pere11e1 m We hhe Mm passes Mruugh Me mm 714 867 06 and 78107 06 Answers to the odd pmblems in section 35 wines in SlopeIntercept form when possible 1 17 8x713 5 yEx77 7 y2x4 9 y8x11 1 y6x710 3 y 2 7 13 x015 y3 17 F7 19 y2 21 y5 23 x6 4 18 27 yx16 29 y x17 31 y7 33 y77x77 1 gr 37 y3x 39 yx 41 y 3 5x7 43 pinks 45 he 47 F73 49 y7x4 Page7 en 3 1 Ll Ch P Road Uh39jr 05 4 k OJ Mnlm r r r CaH39HJ ro I39v 0 9 J Iva ouutj J or r dqu 14 44v Want 6 M p tLJ WW LrobgtJ Chm w J in VS 399 V r Jy 4 64 I39 OLGrc JF U5 f gru vl 6 Kg Lo I4 w l p c P S MTH 60 Simonds class Key Concepts Linear Equations Part 1 MartinGave sections and practice problems 22 13 5 7 9 17 25 odd 22 27 49 odd 23 1 23 odd 31 45 odd 23 25 27 29 47 53 odd Definition A linear equation in one variable x is an equation that can be written in the form ax b c where ab and c are real numbers a i 0 Examgle 1 For each equation either show that it is a linear equation in one variable and state the variable or state why the equation is not a linear equation l11 11 J wc 4 l v Upl llx a 7678t4 b x9 x 0 2 kn lf um lL t gran 1r x c 6x2786 Jo LJ 39nampf BQltOij t J JJLH J I d 0 8214 I y lLc lln39r 01 Coggn LUKArL U 0 6w lquotwlquotJ k no 4 LZJ L7 IrKJ i n l 1 r Page 1 of 8 loaLD MTH 60 Simonds class Definition Two equations are called equivalent equations if they have exactly the same solutions Examigle 2 Decide whether or not each pair of equations is equivalent and explain your reasoning a x717 and x764 L P 39 39 k J k 9 f Ark bfuJpltx i J39 o 15 b t323 andit20 Jri l E 0quot 41 I UKquot f 713 g a yLg J4quot I quot A IALg 1 Lb ao i5 23 LH39 c 97y9and8y28 EQJIUAL JT IJL SJJ 45 439 La 5 0 The addition property of equations Adding the same number or algebraic expression to both sides of an equation results in an equivalent equation NOTE In later classes you will learn that there are exceptions to this rule you do not need to worry about this now however For the types of equations you are solving in this class the statement is true 100 of the time Page 2 of 8 MTH 60 Simonds class Examgle 3 Find The souTion To each equaTion afTer firsT using The addiTion properTy of equaTions To find an equivalenT equaTion of form x k or k xwhere k is a real number Focus on The process We are esTablishing building blocks upon which The remainder of your algebra career will in parT be based x7941 I WWWuwaaddtoxi9 W wtwdrajwytle w vx X q Lfl JalJII 4 2 xf l H 5 7k1 KH 5 50 1 SO l 5 Whatcwwwawbtmotfrovwx wtlwtwdwjmtle w39 vx f 139 jlll u I EJZXT u DTS f 3 L 7 4x53x I WWWuwawlrtraotfrm53x w wtwa rajzwtle 39WWvS LIX 52 Lk J ld h l xrf 5 393 Page 3 of 8 MTH 60 Simonds class 6x18757x WWvaoWVaotfrm6x 18 WWwa Va the w thli 7 WWvaaaddta 75 x WWwo Vay39MStle w vx 7 C x Il 3 7 ll NW1 Cy I7Crr 7X C gar WV l Sgtc 5 7 3 7lr tgtlt 44 93 X okr LK C 13 41 239 00 TL H r4 The multiplication property of equations Multiplying both sides of an equation by the same nonzero number results in an equivalent equation NOTE This property is also true if you divide both sides of an equation by the same nonzero Example 4 Find the solution to each equation after first using the multiplication property of equations to find an equivalent equation of form x k or k x where k is a real number Focus on the process We are establishing building blocks upon which the remainder of your algebra career will in part be based 1 1x35 WWvaowwlL LpZy gx by WWwa VajMytle w vx 5 ix Zf 5 401st 4 s S 39 g y S 7C 2 I76 Page4of8 is rlv xs yr 3 n r 1y 7 35 MTH 60 Simonds class 8 4x I WWWuwawwlt Lply 4x by thwa rajwytle 39w b vx I 114 Nb h i 39 JI l 73 NH yam 5 2 84x I wmwwmmmztx by W xatwa rojmle wbthx q Juld39ku I o K DE 13 quot5 9 T f LY H 7x77 I wmwwwmbpzy 7x by w xatwa rojwytleftwb vx X 7 Lb mwr39w 0 n gtlt 4amp7 xx 39I U 7 X57 Page 5 of 8 MTH 60 Simonds class Solving linear equations in one variable x Completely simplify the expressions on both sides of the equal sign 2 Use the addition property of equations once or twice as necessary so that you have an equivalent equation of form ax b or b ax Wnere a and b are real numbers 3 Use the multiplication property of equations once as necessary so that you have an equivalent equation of form x k or k x Wnere k is a real number 4 Check your solution in the original equation If your solution doesn t check find and fix your mistake State your solution using a complete English sentence Example 5 Solve each equation Focus on The process We are establishing building blocks upon which The remainder of your algebra career will in part be based 8x719735 YxI a Yx quZ X1 L 17 1 4o JJJK 39l a Vx I r 3Y is 1 CJ6J jLLzL li V l 1 14124y lam13 1 j 171L 114941 L 1 l 4 NA 3 l J Page6of8 IJU I 39139 7w7154w u If 9 Id If Y J H 4 3wIY o 3u439f 39 3U IS 7w7154W 7w 1 hJ qd qf39qu IJ 70 2 3u If 2 7 3 5 U 375x727720x 3 2 10 3 S 10 1 17 Joc 4th 34 36 an 3 ISX 3 If 30 57 30 If 7 3 MTH 60 Simonds class CLngg 3a I 3 quot 39 q 39I r LJ 2 3gt Iy Lu MAJWquot a o 19 4 1 43 5qu gt g 4 5 gm a 7u t5 3 u 395 JU 73 CKLDQ 3 3 rz 3 1 ox z 3 J lo i 2397 40 I3 z 3 34 gt jquot stg R 1 Page 7 of 8 MTH 60 Simonds class 6t72t747t73 C L 40484 3 12 5 3 s 5 CLawi W J LLLtK G 4 Lyquot 7 w 9 2xgt 3 4 yEJ Y 4quot 4 34 k Lu 7 3 7 lLgJOIs kv 3 3 ce 1QA v Ue3 N39 3 15 I 5x22715x10x1 1 2 2 Sxll 45X XI 7quot La x 1L loXo if X agt 4393 139H39X mxH h KcLK J 3 7 J 1 gbx quot11quot r3 7H I I 39 Jh 39 2 n JDXHo H 3quot 9 73 C I 3 10 gt0 1ll quot fxrl quot39VXLohm 5 l g Page 8 of 8 Mrh an e Smunus uasse Semam 2 m Wm text The Real Number 39 e and Types of Real Numbers Mus peapTes rs undersfandmg uf numbens neTaves m when we mmhrfype peapTe eau cuummg numbens an e mane furmaw e nmurc numbens These ane The numbens we use m eaum 1 an mane whaTe aeeunnenees uf SW nan ubJecvs We aTsa use Mesa num bens m keep uur yuungruns busy when Mey ve sung une mu many nenenmns anhe anhabev sung Pneny suun anen HHS we genenahy farm an undersfandmg anhe numben zena When we add The uf Numbev wnh a huTe m a mum Numbgys r J 0 1 2 3 4 5 6 7 8 9 10 My Whme Numbevs The Hex Type uf numbens we generaHy eame m uneensvane ane froc uns buv we H pm Mesa my me fur nuw One day m yaun pasv sumeune racked yaun waan wnh The Men uf negmwe num bens A rs Mesa seem yeny svnange e haw ean yau haye a negmwe numben uf ehTehensv The answen uf aaunse Ts Mm yau cam 5a negmwe nu ens eanv make sense when cuummg ehTehens May s why Theyhe nu pan anhe 52 we on cuummg num bens Vuu ean haweyen be abaye an beTaw sea TeyeT If we caH sea TeyeT an eTeyaman uf 22m une easy way fr a m mms abaye sea TeyeT and negmwe yaTues m qus beTaw sea TeyeT If we re measurmg m feev fur exampTe 7 waqu currespund m a pawn mm s 7 fee abaye sea TeyeT whue e 7 waqu currespund m a pmm Mm Ts 7 fee beT The numbens 7 and 77 ane caHed appasne numbens eeaus may ane equTeTsvanv frum 22m and haye appasne sTgns One easy way m Thusvnave HHS Ts wnh a numben hne uw sea eye 7 unus 7 unns 79724757544724 0 1 2 3 4 5 5 7 a 9 negawenumbevs pusmvenumbevs p zem pmm ExaanLL CampTeve each senvenee amenw a The opposne 0742 T5 1 The opposne of 0 T5 0 PageT ans rm suesmunus usssesmnw 2 mwunen T e ser uf numbers cunswsfmg unne narurcu numbers 0 andrne uppusnes unne narurcu numbers 5 caHed M2 ser uf nnegers OK back m frac uns BeneraHy wnen we Mmk uf frac uns we Mmk uf une puswwe nneger dmded by anurner pusnwe mreger We represem new uf a we wnn M2 frachun E 2 represem 2 uur uf 3 yurers wnn M2 freeman E Wemusr as narurm numbers We uppusnes SD 17 7 dB pusnwe fraumns Fur examwe 7 and 77 are uppusne numbers We 4 1 T 2774754 a 9 4Kums 4Xums Any number Man can be wNHen as a quu em uf m y r s caHed a rmuncu number Remember a a frac un comm have a denum mmur uf Zeru Xam 2 3 Shuwfhm M2 numbers 0 8 3 425 and U 3333 are aH rm una numbers I a II Eyery pmm un rne rem number hne currespunds m a umque rem number and eyery rem number ur on Wa urm number Pagez nus MTH 60 Simonds class Section 12 ofyour text Examgle 4 Place each of The following numbers inTo The seTs boxes To which They belong NoTe ThaT mosT of The numbers fiT inTo more Than one seT 15 2s 7 716812 7 714 J3 J25 474 0 922124457 Law a 72 7 nuier Rational numbers Irrational numbers lr ILJIV 39ILl I f I 3 r 0 71m Hr Integers L f 1l ld o Whole numbers ago 1 Natural numbers Counting numbers ET2m Page 3 of 6 MTH 60 Simonds class Section 12 ofyour text Examgle 5 Decide wheTher each sTaTemenT is True or false If The sTaTemenT is false give an example ThaT shows why The sTaTemenT is false 0 Every inTeger is a naTural number V 1 nVn 6J1 IL 60 I 7 LJ kb quot 1 I L4 0 Every whole number is a raTional number k 4g 0 Some naTural numbers are irraTional numbers N 3 t u lt 39 Egualiy and Inequalities As previous menTioned every real number corresponds To a unique poinT on The real number line When ploTTing Two differem numbers on a TradiTional number line one of The Two numbers lies To The lefT of The oTher number The number on The lefT is called The lesser of The Two numbers and The number on The righT is called The greaTer of The Two number a lt b means ThaT a lies To The lefT of b along The real number line The expression a lt bquot is read aloud as a is less Than b a gt b means ThaT 1 lies To The righT of b along The real number line The expression a gt b quot is read aloud as a is greaTer Than b a b means ThaT a and b are in facT The same poinT along The real number line The expression a bquot is read aloud as a equals b a g b means ThaT a lt b or a b The expression a g b quot is read aloud as a is less Than or equal To b a 2 b means ThaT a gt b or a b The expression a 2 b quot is read aloud as a is greaTer Than or equal To b a lt b a gt b a S b and a 2 b are called inequaliTies or inequaliTy sTaTemenTs a b is called an equaTion Page 4 of 6 MTH E rswmunds assrsemun1 2 uWuuHEM EXamg E E Indwcavz rough y thz posmon of each of thz foHowmg numbers on thz pmwdzd number w my hm 2cm 7 appm r2 Sta2 and than answer vhz subszquzm quzsnohs LIS IYV propzr mathemancm symbo s and Hammer I quot fj II 3 I quot I J 11 8 Ir 747 23114 note that through 5 dngs 11410 appraxwmate yequa s Y E J r5 W 1 1 d L U I t 3 1 J IIJ Whmh s the true statement A 2 s 2ss thah4 0M 3 2ss than 11 3 3 lt 4 3 Whmh s thz mg statement 7s 235 thaw or 4 s 233 Man 713 17 ll quot q lt 3 Whmh s the ma statement 114 7r gt114 T 51 Whmh s thz mg statement 4114 in gt114 TT lt3H or Irlt3147 or 77rlt3147 11 11 Whmhwstruz gsrnmrgzrnv H L Whmh 3 true 2 V V 4 V 1 Au LL11 139 TI 3 L J Page 5 uf a Absolute Value MTH 60 Simonds class Section 12 ofyour text The symbols lnl are read aloud as The absoluTe value of nquot The absoluTe value of The real number n is The disTance beTween ThaT number and The number zero along The real number line FacT 1 The absoufe value of a number is never negafVe FacT 2 OpposiTe numbers have equal absoluTe values Examgle 8 For each given inequaliTy or sTaTemenT sTaTe wheTher The sTaTemenT is True or is false 799 l 71022 71022759 59 Examgle 9 Simplify each expression 3779 3 l39 l 3 C 71022759 979 71022 59 71022759 r 7 775l8778 799 l 7102259 7102 2759 p 7 Y39I qll T7lSquotgq Page 6 of 6 7Y 7 510 70 397 The Real Numbers Rational Numbers Integers 1 Whole Numbers 0 Natural Numbers 7 6 2 L 7L llm l f 7 Irrational Numbers K I l 3 a orms a forever repeating pattern Integer Any number that simpli es to a number in the list Whole number Any number that simpli es to a number in the list Natural number Any number that simpli es to a number in the list The real numbers The set ofnumbers that correspond to points on the real number line 7473727101234 01234 1234 Rational number Any number that c n be written as a 39action of integers ln decimal form a rational number either terminates or f Irrational number Any real number that is nota rational number If is an irrational number The square root ofa natural number is either itselfa natural number like I49 or is an irrational number like 415 112 MTH 60 Simonds class Key Concepts Properties of real numbers Language of algebraic expressions rtinGave sections and practice problems all odds Ma 18 21 1 65 odd 83 85 87 5 Properties of real numbers AJ 6 J n H An The associative prope es The commutative properties The distributive property abcabc abba abcab ac ab39cabc a39bb39a L c L KL mulL39plicalx39h Jillrst 3Hquot h M Example l f J J 5 Fch A39 Simplify each expression after first applying the associative andor commutative properties Write your solutions in the lined up equal signsquot format Note An algebraic expression is not simplified so long as there are numbers remaining that could be addedsubtractedmuItipliecl or divided Simplify 8 7 x Simplify 7 967 7 9 Yltlx K1lx 7x 7 1 2 M fw i w twig wlm H4 A ip KHv ford nalDc 7 PM O llt 7s a 3yx ginw Ciel C Simplify 374x Simplify 6 ll y I 1g Anewquot r39wty ll J calle LL39Prual Cowvi bk 7 N H l 1 I mvlkrliLv 39 T Lc39r39ww Aquot 0 le Ana5 Page 1 of 12 212 Examgle 2 Simplify using The properTies of real numbers WriTe your soluTions in The lined up equal signsquot formaT and for each line sTaTe The properTy used Simplify 3x 9 m d lvilu hdk 3 x 4 H r quot J Simplify4x78 Gonna149 L i quotrim I L4H 4M t 2 W 76 Jllpwh l l 3quotle Zm 3 Q 4 7L Simplify 712b743 D I i r lzL Y f L3 Ag M Hmw qu H 39Il L v7 312 MTH 60 Simonds class Examgle 3 Use The disTribuTive properTy To wriTe each expression as a producT 9x9y Z 7tc7w 7 39E J 3x12 Identities and Inverses Zero is called The addiTive idenTiTy a 0 0 a a One is called The mulTiplicaTive idenTiTy a11a a a quot and 7a quot are called addiTive inverses a 7a 7a a 0 1 1 a quot and quot are called mulTiplicaTive inverses a i 0 a a 1 a a a Exalee 4 WhaT is The reciprocal of The mulTiplicaTive inverse of 5 WhaT is The sum of The addiTive idenTiTy and The addiTive inverse of 78 2 WhaT is The producT of The mulTiplicaTive inverse of 77 and The mulTiplicaTive idenTiTy WhaT is The addiTive inverse of 0 WhaT is The mulTiplicaTive inverse of 0 Page 3 of 12 412 MTH 60 Simonds class Language of algebraic expressions Algebraic Terms The product of numbers and variables raised to powers is called an algebraic term This is an extremely simplified definition but serves our purposes for now When all of the numeric factors of a term have been multiplied the resultant number is called the coefficient of the term In simplified form the coefficient of the term is always written first and the variables follow in alphabetical order 9 Two terms that have exactly the same variables raised to exactly the same powers are called common or like terms Example 5 What is the coefficient of the term 3x 76 EmC quot7 LL ngm E Y 1 Write the term 77y3 77sz in simplified form and state the coefficient of the term 3 39l 139 Y 392 733 1 37 J lx J I g 32 lku Lot llu l 5 l L uk Aha U4k Lu fxaty o k u n O o r l Which terms are like terms 3xZ ys 7 2x5 yz iys x2 What is the coefficient of iys xi 0 JL g 49m 3xxj 3 ij 0 J XL 0V1 39b be M 39ln o I r JJ M s L L gc39u M E x Page4of12 S I 512 MTH 60 Simonds class Example 6 Simplify 3x 5x 3x ms 53 f e 3X gt 2lt37X1X In essence what did we add when we simplified 3x 5x Con FVJ II39KL DAGLLJ 4 1 lAg LJJ It What property of real numbers justifies the process A LJ Lac THIH JK OkLltwamprJJ Simplifying algebraic expressions There are many properties to a simplified algebraic expression Three properties are that the expression contains no grouping symbols all like terms have been combined and all terms are written in simplified form Example 7 a Marcia and Jan were having a dispute Marcia said 7 3x and 10x are the same thingquot and Jan said Marcia Marcia Marcia you are so wrongquot Evaluate each of the expressions when x 2 to establish which of the sisters was correct UkLx X72 391 3 737 2144 3 JN J rLjL l l 7 0J 3X a rl v dquotlllltC I n1 b U l 71L Chubl Gorrr d ly J Arc 15 0lt 0 l 791 0 Page 5 of 12 91va KJLE S 5 JfIVJ 0Y 5 v5er 54 3 6fL 6 Low H4MTH60 Simt s class l Lo cam 14 L Un ntJ b Willis and Arnold were rapping abouT maTh WiH i said 10x2 7 3x is equivalenT To 7x1quot and Arnold said WhachuTalkingabouT Willisquot EvaluaTe each of The expressions when x 3 To esTablish which of The broThers was correcT vx Z L JR K 3 L 1 XL 397 3 oxL3xo3l 3 l 3974 4 33 63 f0 7 Yl Larry and GilberT each had a collecTion of xs and sTicks Larry had 8 xs and 5 sTicks in his collecTion and GilberT had 4xs and 12 sTicks in his H ll Afnal l Pi C If The Two lads combined Their collecTions how many xs and how many sTicks would They have Km xxxx 2 0 nJ HH iw xxxx H l l39 Ix WW 7 d WhaT is The correcT simplificaTion of 8x 5 4x12 Yx is lt4x m lt7 gtgtlt My LC 47 Eddie and his grandpa each had collecTions of one headed and Two headed baTs Eddie had 4 one headed baTs and 9 Two headed baTs Grandpa had 16 one headed baTs and 49 Two headed baTs 8 If The Two MunsTers combined Their collecTions how many one headed baTs and how many Two headed baTs would They have 20 ow Lach barb 4m bylaw QOXTTY f WhaT is The correcT simplificaTion of 4x 9x2 16x 49x2 sz 4136quot if Mac H x Minx wm xquot 1 Q 0 x 5 Page 6 of 12 612 MTH 60 Simonds class Exalee 8 When simplifying algebraic expressions you can add or subtract like terms into a single term There is no way to add or subtract unlike terms into a single term Write The simplification of each expression into The provided blank 8x5710x V l 9 H 0 g fix 976t18 12x8719x2 7X 0 10 7 45x2 73x2 1x 44 496 5x2 73x2 JxL Jrle 4x5x273x 30quot gtlt 45x273x 4 Jr Sat3 XL Xx I T 4y78x710y 2y I 39il aw 391 ltlt Page 7 of 12 1 1 5 3 x78 x7 2 6 5 393y 1 h 712 812 MTH 60 Simonds class Example 9 When a negative sign is present in front of an expression in parentheses the sign on each term in the parentheses changes when the parentheses are removed a Distribute each negative sign and write the result in the provided blanks x 77x7 y8 7H 76x2 476 Syzti y 775y29y 5y2 9y s5yzs9y Page 8 of 12 912 MTH 60 Simonds class b Dis rribu re each subTracTion sign and Then simplify The resul r 57974x 10x77712x 12yy20 718x718x6x2 277100b2 746t7t717 67710w78 7 719x7719x6 7 79xx2 79x6 5x7719x673 Page 9 of 12 MTH 60 Simonds class Problem 1 Comple rely simplify each expression Make sure Tha r you presen r your work in The proper forma r Simplify 2x 7 5 Simplify 3y7 7 4y7 2 Simplify 8517 67 d Simplify 0180x2 7 90x7 14x7 6x2 Page 10 of12 MTH 60 Simonds class Problem 2 CompleTe each senTence wiTh one of The wordsphrasesnumbersnames below Pick The one ThaT makes The senTence True numerator difference denominator positive Inverse opposite Huffy ostrich negative P If you mulTiply 87 wiTh iTs muTipicaTive inverse The resuT is b 34734 37 is an exampleof The 2 c 72 Is 0 When reading aloud 776 The firsT minus sign is read as and The second minus sign is read as e 69878769isanexampleofThe f If you mulTiply 87 wiTh The addiTive idenTiTy The resuT is g 1279199 isanexampleof h If The opposiTe of lxl is noT zero Then iT is definiTely If you mulTiply 87 wiTh The muTipicaTive idenTiTy The resuT is j 8 5 75 8 0 is an unconscious applicaTion of The Page 11 of12 MTH 60 Simonds class Problem 3 Decide wheTher each sTaTemenT is True or false T or F No maTTer whaT values x y and z assume x 7 yz x2 7 yz TorF 7747877478 T or F 4x3 and 3x4 are like Terms T or F If you mulTiply a number by The mulTiplicaTive idenTiTy The resulT is always The original number T or F No maTTer whaT values x y and z assume 6 y Z x y Z T or F No maTTer whaT values x y and z assume Ix y z x ly 2 T or F If you add a number To iTs addiTive inverse The resulT is always The original number Problem 4 Name ThaT numberexpression rLr39reorhsvmoprp WhaT number do you add To 74 so ThaT The resulT is 0 WhaT number do you add To 83 so ThaT The resulT is 0 WhaT number do you mulTiply wiTh i so ThaT The resulT is 1 WhaT number do you mulTiply wiTh 76 so ThaT The resulT is 1 WhaT number do you subTracT from To 162 so ThaT The resulT is 0 WhaT number do you divide 74322 by so ThaT The resulT is 1 WhaT would you subTracT from 3x so ThaT The resulT is 0 WhaT would you add To 6 so ThaT The resulT is 0 WhaT would you add To 7254x2 so ThaT The resulT is 0 WhaT is The soluTion To The equaTion x 11 87 WhaT is The soluTion To The equaTion 5x 35 WhaT is The soluTion To The equaTion 10x 11 22x 5 Page 12 of12

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